Combining Masculinizing Resistance, Rotation, and Biocontrol to Achieve Durable Suppression of the Potato Pale Cyst Nematode: A Model

ABSTRACT The pale cyst nematode, Globodera pallida, is a pest that poses a significant threat to potato crops worldwide. The most effective chemical nematicides are toxic to nontarget organisms and are now banned. Alternative control methods are therefore required. Crop rotation and biological control methods have limitations for effectively managing nematodes. The use of genetically resistant cultivars is a promising alternative, but nematode populations evolve, and virulent mutants can break resistance after just a few years. Masculinizing resistances, preventing avirulent nematodes from producing females, might be more durable than blocking resistances, preventing infection. Our demo‐genetic model, tracking both nematode population densities and virulence allele frequencies, shows that virulence against masculinizing resistance may not be fixed in the pest population under realistic agricultural conditions. Avirulence may persist despite the uniform use of resistance. This is because avirulent male nematodes may transmit avirulent alleles to their progeny by mating with virulent females. Additionally, because avirulent nematodes do not produce females themselves, they weaken the reproductive rate of the nematode population, leading to a reduction in its density by at least 20%. This avirulence load can even lead to the collapse of the nematode population in theory. Overall, our model showed that combining masculinizing resistance, rotation, and biocontrol may achieve durable suppression of G. pallida in a reasonable time frame. Our work is supported by an online interactive interface allowing users (i.e., growers, plant health authorities, researchers) to test their own control combinations.

Avirulent larvae (genotypes AA and Aa) differentiate into adults with a proportion m A (k) of males, and a proportion 1 − m A (k) of females.Virulent larvae (genotype aa) differentiate into adults with a proportion of m a (k) of males, and a proportion 1 − m a (k) of females.
Sex allocation depends on host resistance during season k: in which 0 < m < 1 is the male fraction when masculinizing resistance does not occur or is broken.The transition from the larval stage to the adult stage is given by the following equation: M aa (t k + τ 1 ) = sm a (k)J Aa (t k ) . (3) 3. We assume that all adults survive until mating occurs (time t k + τ 2 ).We assume that males and females are homogeneously mixed, and that mating occurs at random.The next three steps lead to the female-to-egg transition from females to eggs.
(i) We first derive the frequencies of allele A and a in males and females.The variables M = M AA + M Aa + M aa and F = F AA + F Aa + F aa are the total densities of males and females, respectively.The frequency of allele A in females, the frequency of allele a in females, the frequency of allele A in males, and the frequency of allele a in males, are respectively: (ii) The allelic frequencies yield the following next-generation genotypic frequencies: (iii) Egg density is then obtained, for each genotype, as the product of the total female density, the average number of eggs per female, e, and the next-generation genotype frequency from equation ( 5): Remark.Equation ( 6) encompasses polyandry, since males have equal access to females.
4. A fraction (1−µ)(1−h) of eggs naturally survives a single period of host absence.This fraction is the product of the average fraction of viable eggs (1 − µ) with the fraction of cysts that did not accidentally hatch (1 − h).
In addition to mortality, a fraction (1 − b) of nematodes survives biocontrol application (b is the biocontrol efficacy fraction).Not using biocontrol amounts to setting b = 0.
The rotation number r is the number of years without growing potatoes.We assume that biocontrol is applied every year, regardless of whether potato is grown or not.
As the potato growing season is relatively short on an annual scale (16 to 18 weeks), we assume, for simplicity, that annual cyst mortality is the same regardless of whether the host absence period is one year or less (one year minus the growing season).
At the beginning of the next season, i.e. at time t k+1 , we assume that all eggs hatch from cysts.Therefore, the nematode generations do not overlap.
The transition from eggs to larvae is thus given by: To simplify notations, we introduce the reproduction number R ′ , accounting for biocontrol and rotation: We also introduce the female fractions

S1.2 Compact model
Renaming the larvae genotype densities we summarize equations (3-7) in the following discrete-time dynamical system: We modify model (10) to account for intraspecific competition among larvae to host access, in a Beverton-Holt form: in which c is an intraspecific competition parameter.System (11) defines our demo-genetic model of potato cyst nematode population dynamics.

S1.3 Basic demographic model
If only susceptible plant hosts are grown over years, the male fractions are m A (k) = m a (k) = m and the female fractions are f A (k) = f a (k) = (1 − m), for all k ≥ 0. Model (11) simplifies as: Let N k = X k + Y k + Z k be the total nematode density.System (12) yields: This is the classical Beverton-Holt model.It has at most two non-negative equilibria: N = 0, and N = ((1 − m)R ′ − 1)/c (if and only if R ′ > 1).The nematode-free equilibrium, N = 0, is stable , the nematode population grows until reaching its carrying, Next we show that genotype frequencies stabilize at the Hardy-Weinberg equilibrium.Using equations ( 12) and ( 13) yields the following genotype frequencies: The frequency of allele a on season k is Using equation ( 14), one can check that a k+1 /a k = 1 for all k ≥ 0. Therefore, a k = a 0 for all k > 0. The Hardy-Weinberg equilibrium follows: for all k > 0,

S1.4 Demo-genetic model with masculinizing resistance
In this section, we consider masculinizing resistance to be the only potato variety grown over years.
The male fractions are therefore m A (k) = 1, m a (k) = m for all growing seasons k ≥ 0; consequently, the female fractions are f A (k) = 0 and f a (k) = (1 − m) for all k ≥ 0.
Equation ( 11) yields X k+1 = 0 for all k ≥ 0, meaning that the homozygous avirulent genotype (AA) cannot persist.Therefore, N k = Y k + Z k for all k > 0. Equation ( 11) simplifies as: Summing both equations above yields: Let the frequency of the virulent genotype (aa) on season k be We obtain 15) can therefore be equivalently expressed as a demo-genetic model coupling population and virulence dynamics: . (16)

Non-dimensionalization
We rescale model ( 16) by letting n k = cN k , which yields:

Stability of equilibria
The stability of a given fixed point (n, v) depends on the spectral radius of the following matrix: .
Table 1: Fixed points of the demo-genetic model ( 17) and their stability.

S1.5 Time to effective suppression
The nematode pest can be considered as effectively suppressed from growing season k † if its density, N k , does not exceed a certain acceptance threshold, τ , for all k ≥ k † .We are interested in deriving the length of time required to achieve effective suppression, k † (r + 1).For simplicity, we consider that the frequency of the virulent genotype, v k , is initially at equilibrium: i.e., for all k ≥ 0, v k = v ⋆ , as defined in equation ( 18).Model ( 16) simplifies as: We assume N 0 > τ (the nematode population density is initially above the acceptance threshold).
We next focus on dynamics leading the pest to extinction, which occur if and only if (1 − m)Rv ⋆ < 1, or equivalently R < 2. Using the explicit solution of equation ( 20), that is we derive the generation k † from which, for all k ≥ k † , N k < τ : the latter inequality is equivalent to Hence, This graphic also illustrates the decrease of biocontrol efficacy needed as e diminishes.
Figure S3: Time required to decrease nematode density under the acceptance threshold τ = 1 nematode per gram of soil, with biocontrol efficacy fraction b = .65and default parameter values, for a range of rotation numbers: r = 0, 1, 2, 3, 4. For high rotation numbers (r = 3, 4), the masculinizing resistance does not particularly speed up nematode suppression (initially at virulence frequency v = v ⋆ ).By contrast, for r = 2, growing a susceptible variety does not achieve suppression, whereas masculinizing resistance does.Lower rotation numbers (r = 0, 1) do not achieve suppression.

Figure S2 :
Figure S2: Rotation number required for long-term suppression of G. pallida under masculinizing resistance as a function of the biocontrol efficacy b for different values of the number of eggs per cyst, e.The minimum rotation number is maximized for b = 0, and is a decreasing function of e.This graphic also illustrates the decrease of biocontrol efficacy needed as e diminishes.

Table 1
summarizes the stability of the fixed points.